# Differential forms as functionals on curves

Please give me a reference to a book or lecture notes where the following stuff is studied.

Let $M$ be a Riemann surface with boundary $\partial M$ (but not necessarily, any smooth $n$-dimensional manifold suffices if the following statements make sense in this case).

1. What does it mean for $1$-form $\omega$ on $M$ to have $\int\limits_\gamma \omega = 0$ $(\operatorname{mod} 2\pi)$ for any closed loop $\gamma$?
2. If $\omega_1$, $\ldots$, $\omega_N$ are closed forms which form a basis of $H^1(M)$ (de-Rham cohomology) then there are non-homotopically equivalent loops $\gamma_1$, $\ldots$, $\gamma_N$ in $M$ such that $\int_{\gamma_i} \omega_j = \delta_{ij}$. How to prove this? Is it possible to say more about these $\gamma_1$, $\ldots$, $\gamma_N$? Do they have some interpretation on terms of $H_1(M)$ (singular homology)?
3. As in 2., if $\omega_1$, $\ldots$, $\omega_N$ are closed 1-forms which form a basis of $H^1(M,\partial M)$ there exist non-homotopically equivalent loops $\gamma_1$, $\ldots$, $\gamma_N$, non-homotopically equivalent to any component of $\partial M$, such that $\int_{\gamma_i} \omega_j = \delta_{ij}$. What can we say about these curves in comparison with question 2?
• I think that do Carmo's book "differential forms and applications" will be helpful for problem 1 – HK Lee Nov 26 '13 at 10:51
• @HeeKwonLee unfortunately I haven't found the answer in this book, do you know the answer? – Appliqué Nov 26 '13 at 17:05
• Sorry i did not read condition well – HK Lee Nov 27 '13 at 1:07

What you are struggling with is called Poincare Duality for manifolds with boundary. If $M$ is a compact oriented $n$-dimensional manifold (in your case, $n=2$, $k=1$), the Poincare duality reads: $$H^k(M)\cong H_{n-k}(M, \partial M), H_k(M)\cong H^{n-k}(M, \partial M).$$ In the case $n=2$, you also have the Kronecker duality $$H^k(M)\cong (H_k(M))^*, H^k(M, \partial)\cong (H_k(M,\partial))^*$$ (in general, you have to use real coefficients or use the Universal Coefficients Theorem, which is more subtle). The best place to read about this that I know is the book